It might be a terrible idea, but what if every game had a pre-defined mode that is either set (for everyone to know) or random (so no-one knows). For example, let's just compare the _beliefs_ of young-Earth creationists and secular humanists (depending on your definition, they might both be religions, either might be, or neither is). Both have worldviews that makes claims about science (one saying a God of order made everything - recently - whereas the other is saying there is nothing supernatural and everything that exists does so by natural causes, and came into existacne that way).
If there is a preset mode (or age of the universe), then there would be a science bonus to a civ that picks the right one. There could be factors built into the game that change because of this, such as the amount of comets, the apparent age of stars, ruins and the development of NPC players (we'd expect an older universe to have older civs - give 'em a head start, just not too much). Those factors would also be hints of which view is correct in that universe.
As for details on religion... if there indeed is a God as universal as any ONE GOD must be, then all civs would likely have some cultural trace of GODs interaction with that particular civ, and be able to pick that religion. It may, however, have changed over time, things been added and reformed. Just compare the Abrahamic religions and their very varying views and practices. Actually, you can compare a number of christians from different denominations and you'll end up with very varying views.
A simple tolerance slider could be good enough, and that would be influenced by so many other things.
Consider this idea: how would the planet's scientific community respond to a technologically superior civ that are radical young-Earth (or young-Generica) creationists? Would our scientists adopt their views? Would we enforce our views on them? What would happen?
We might even be invaded by crusaders from outer space, aliens with no tolerance for any view but their own. What would we do then?
Essentially, there's the need for two meters, tolerance, and cultural diversity which would include religion. I suspect any culturally homogenous civ would suffer scientific setbacks as the one true science is academic law. On the other hand, we'd have all sorts of minor cultural clashes is diversity is too high. Every other european country has some religious, philosophical, or cultural hatred for another, not to mention the conflict between "Islamic" terrorists and the "Christian" west.
Picking a state religion would start to homogenize the civ and unpicking it would diversify the civ. Homogenous means less corrpution, diverse means faster science. There only a need for two choices but these might be different for most players, some having things in common, some not.
If Earth had a religion called CNN and Genericans of religion CBS came, they have factor C in common, and both have palced it first. A similar system could be devised, only you'd need to interact with the other civ to learn of their culture and religion/philosophy. With the C in common, they could have an easy time dealing with each other.
Naming the religions is hardly necessary, but tagging them like I did above, only with features rather than letters, could be good. Each feature could then affect the civ as a whole as well. Core values, some of which can be combined, might work.
Believe or die (war morale bonus, but makes population unrestful when populations are diverse or when at peace with civ with very different beliefs)
You're accountable after death (reduced corruption, but some medical problems (through fear of death) may arise)
Respect people (increased tolerance and morale but reduced... production, maybe? plus reduced war morale)
Sabbath (decreased production, increased health)
Recent creation (see suggestion waay way above, incompatible with All natural universe)
Wealth is key (increased trade incomes, increased corrpution)
All natural universe (see suggestion waay way above, incompatible with Recent creation)
"Human" sacrifices (increased health (getting rid of the old, and sick), decreased corrpution (yeah, let's kill the criminals while we're at it), decreased tolerance) (probably incompatible with respect ppl*)
Plus many many more
*) what if they only sacrifice volunteers?
Christianity, or whatever branch of it that I best fit into, would be: recent creation (Creation), respect people (Respect), and you're accountable after death (Accountable)... in some order. Probably ARC or ACR. If I'd make that the belief of my civ, I'd have increased tolerance and morale, possibly science as well, and reduced corrpution, but also reduced production, morale during war, and some health problems.
An alien civ with most of those on their top three priorities list would probably be a likely ally, while my civ would have trouble dealing with one that puts wealth above accountability and picks naturalism over supernaturalism. Even more so if they've got believe or die somewhere relatively high.
Maybe there could be a ranking system for these, one that measures what the ppl have, one that you set and gradually influence them to fit into. Too drastic changes would cause some unhappiness, maybe even unrest.
The good thing about this latter idea is that it stays away from the debateable details about God, gods, the universal divine, or the lack thereof. Tangible gameplay effects, and not any whining about whose real-life religion, or lack thereof, is the ultimate truth.
I'm tempted to end this with "besides, my religion can kick your religion's intangible ass any day of the week".
I'm not gonna.